Elements of algebra

attach to our definition is this, that there is some number or interminable fraction which, when multiplied into itself, produces a. Now, if a be a negative quantity, it is obvious that no number or fraction, positive or negative, of any kind, will express its square root; for the square of every possible quantity is positive. That which does express it, therefore, which is only the indication of an operation, has been designated an impossible quantity, or an imaginary quantity. When it appears in the final result of an operation destined to furnish a numerical conclusion, the designation of impossible is appropriate: in such cases, the conditions of the problem cannot be fulfilled; but it is not to cases of this kind that the appearance of this symbol is confined. It enters largely into the most important operations of algebra, and affords facilities for computation by connecting together different forms.

The following is the mode of its introduction, (a +√√— 6a) (a−√√ b2) = a2 + b2.

Here we have performed a multiplication in the following manner, a × √-b2 is supposed to be written a√, as though it were the product of two numbers: and -. -bis, from the definition of a square root, equivalent to b2; our definition in this case being, that the square root of an expression is such an expression, that, when combined with itself according to the form of multiplication, it produces the former expression.

+72. Instead of √b2, it is most common to adopt the notation b√1, a result which is arrived at, by supposing that the symbol √-1 has such a value as to admit of the application to it of the laws given in Art. 29. We shall then suppose 1 to be a factor similar to other factors, conceiving that b-1 signifies the repetition of the quantity √1, 6 times: by this supposition, the application of this symbol is made. strictly arithmetical. For instance, if we multiply b√-1 by c, we proceed according to the usual rules for multiplication, being satisfied that the subsequent process of again multiplying

the result by 1 will give an interpretation to that which is not as yet susceptible of one: we should write

c(a+b√-1)=ca + cb √√1,

from the knowledge that, if we multiply the second side of this equation by a-b-1 according to the same rules, we should obtain

ca2 + abc1-abc √√- 1 + cb2

CHAPTER IV.

ON FRACTIONS.

73. The notion of an algebraic fraction is so intimately connected with that of an arithmetical one, that we can scarcely define the latter without referring to the former. We therefore commence with an arithmetical fraction as the most simple method of proceeding.

An expression such as

is termed a fraction: if a and b b

are numbers, the expression denotes that one unit (as one foot) is divided into b equal parts, of which a are expressed by the fraction: thus, if a = 2, b = 3, expresses 8 inches, by denoting that 1 foot has been divided into three equal parts and two of them have been kept.

Another mode of viewing this subject presents itself thus: It is obvious that the fraction where a and b are whole numbers,

may be regarded as signifying that a units are divided into b equal parts, of which one division is represented. By taking b such fractions, we shall therefore obtain a units. In other words,

is such a quantity that, when multiplied by b, the product is a, and is therefore equivalent to a divided by b. If we retain this idea as our fundamental one, the notion which we shall attach to

in other cases is this, that it represents a magnitude which, being multiplied by b produces a. Hence we have the following DEFINITION :-A fraction is such, that when multiplied by the denominator, the product is the numerator.

REDUCTION.

The fundamental proposition of reduction of fractions is this:

74. PROP. If the numerator and denominator of a fraction be multiplied by the same quantity, the fraction is unaltered in value.

Let be the fraction, where a and b are integers, then b